These models are actively tested and refined until they match the observed field to within the limits set by natural and unavoidable experimental uncertainties.
It is very important to keep in mind that the empirical model is built entirely from a statistical analysis of the data themselves.
It is essentially independent from whatever physics might be involved in generating the field.
One of Barnes' many mistakes is to insist that only the dipole component of the magnetic field is generated by currents within the Earth, and that all lesser components (called 'higher order components' in physics jargon) are generated by some other process, such as magnetic rocks or telluric currents (electric currents induced in the crust, for example, by lightning in thunderstorms, or induced as a reaction to currents in the ionosphere).
This mathematical tool was invented by the ubiquitous German mathematician, Carl Freidrich Gauss, circa 1835, for the purpose of evaluating the Earth's magnetic field.
This ingenious method uses an infinite sum of trigonometric functions to evaluate a field, on the surface of a sphere embedded in the field.
Simply stated, it has been shown that turbulent motions within an electrically conductive fluid will generate magnetic fields.
This is a poor idea, as it is very hard to reconcile with the spatial extent of these higher order components, as illustrated by figure 2.5 in [1, page 25].
It is very hard to imagine a field of magnetic rocks, or a coherent telluric current, either of which is as large as one half or one quarter of the Earth.
But Barnes is essentially forced to commit this error as a natural result of his rejection of dynamo theory, and his model of an exponentially decaying current in the Earth's core as the source of the Earth's magnetic field.
I make note of this error in order to emphasize that Barnes' failure has a lot to do with very fundamental aspects of the problem, which he hides behind a smoke screen of superfluous detail, as we shall see.